Ok, back to the original question. I think I need to do some slight fixin' on it.

FrediFizzx wrote:FrediFizzx wrote:This expression seems a bit odd to me.

In order to get the probabilities for each of the four outcome pairs say in a large simulation, they first have to be averaged over many trials per (a-b) angle. It seems to me that in a proper simulation each of the four probabilities are going to converge to 1/4 for very large number of trials. At least that is what I am finding with our latest simulation.

Ave ++ = 0.248903

Ave -- = 0.248803

Ave +- = 0.246508

Ave -+ = 0.255786

That was for 10,000 trials. For 5 million trials,

Ave ++ = 0.249787

Ave -- = 0.249991

Ave +- = 0.250293

Ave -+ = 0.249929

Much closer to 1/4 each. So, for analytical purposes, it doesn't seem unreasonable to assign 1/4 to each of the four outcome pair probabilities.

Ok, now for the next part of this.

QM assigns for those 4 outcome probabilities,

Again, in a simulation with many trials, we have to average

and

over all the (a-b) angles. Lo and behold, when we do that we obtain,

,

,

Because

.

So, it seems to me that all of the parts of the original E(a, b) expression are all equal to 1/4. Analytically-wise.

Ok, now a question. Since all P(++)'s, etc. are equal to a 1/4 and the average of

= 1/4, etc., does that prove that

, etc. for our analytical situation? Keeping mind that P(++) and

, etc. are actually averages.

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Ok, what we actually have here for the simulation is

.

So, the question is actually this. Does,

??

Now the only way to equate those two would be by their totals or length of the lists. So, let's see. For a million trials, there are 250,000 events that are (++). and there are 250,000 events that are

which makes sense since basically there would be a 1/4 each of the sin^2 and cos^2 probabilities. So, they match quantity-wise. But what we are actually trying to find out is this,

. Is that true for our simulation? Now, we know that the probability of getting (++) for the simulation is 1/4 so

Now, we also know the average of getting,

So,

Then by virtue of what QM says about it, I think we can say that,

for both the simulation and the analytical formulas. But maybe more proof is needed?

You can send me a PM if any questions or comments.

.